Indefinite integral

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Bill Foster

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Bill Foster 55 years, 4 months ago

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Name	Status	Author	Last Modified
Indefinite integral	draft	Bill Foster	06/08/2020 08:41
Julie's copy of Indefinite integral	draft	Julie Crowley	06/02/2017 15:33
Simon's copy of Julie's copy of Indefinite integral	draft	Simon Thomas	21/03/2019 10:24
Indefinite integral 2 (custom feedback)	draft	Clodagh Carroll	09/03/2020 11:10
Jo-Ann's Indefinite integrals	draft	Jo-Ann Lyons	17/04/2019 13:24
Jo-Ann's copy of Indefinite integral	draft	Jo-Ann Lyons	17/04/2019 14:15
Indefinite integral 2	Ready to use	Jo-Ann Lyons	24/04/2019 15:29
Indefinite integral 2	Ready to use	Jo-Ann Lyons	24/04/2019 15:31
Indefinite integral 2	draft	Xiaodan Leng	10/07/2019 21:57
Indefinite integral 2	draft	Xiaodan Leng	10/07/2019 21:58

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Question statement

Find the following indefinite integral.

Input the constant of integration as $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
b	integer	2
d	integer	9
n	integer	4

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Generated value: `integer`

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$\displaystyle \int \simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \phantom{{}}$ Unnamed gap

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$ .

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Let $y = \simplify[std]{{a}*x+{d}}$ . Then,

$\simplify[std]{{b}/(({a}*x+{d})^{n})} = \simplify[std]{{b}/(y^{n})}.$

Now,

$\int \simplify[std]{{b}/({a}*x+{d})^{n}} dx = \int \simplify[std]{{b}/(y^{n})} \frac{dx}{dy} dy.$

Rearrange $y = \simplify[std]{{a}x+{d}}$ to get $\displaystyle x = \simplify[std]{(y-{b})/{a}}$ , and hence $\displaystyle\frac{dx}{dy} = \frac{1}{\var{a}}$ .

$\displaystyle \int \frac{1}{y^n} dx = -\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:

$\int \simplify[std]{{b}/(y^{n})} \frac{dx}{dy} dy = \simplify[std]{{b}/({n-1}*y^{n-1})} \cdot \frac{1}{\var{a}} + C = \simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.$

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Bill Foster 55 years, 4 months ago

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Generated value: `integer`

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Indefinite integral

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`